Trigonometry Examples

$\cos (22.5)$

Step 1

To convert degrees to radians, multiply by $\frac{π}{180 °}$ , since a full circle is $360 °$ or $2 π$ radians.

Step 2

The exact value of $\cos (22.5)$ is $\frac{\sqrt{2 + \sqrt{2}}}{2}$ .

Tap for more steps...

Step 2.1

Rewrite $22.5$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\cos (\frac{45}{2}) \cdot \frac{π}{180}$ radians

Step 2.2

Apply the cosine half-angle identity $\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

$(\pm \sqrt{\frac{1 + \cos (45)}{2}}) \cdot \frac{π}{180}$ radians

Step 2.3

Change the $\pm$ to $+$ because cosine is positive in the first quadrant.

$\sqrt{\frac{1 + \cos (45)}{2}} \cdot \frac{π}{180}$ radians

Step 2.4

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$\sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}} \cdot \frac{π}{180}$ radians

Step 2.5

Simplify $\sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}$ .

Tap for more steps...

Step 2.5.1

Write $1$ as a fraction with a common denominator.

$\sqrt{\frac{\frac{2}{2} + \frac{\sqrt{2}}{2}}{2}} \cdot \frac{π}{180}$ radians

Step 2.5.2

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{2}} \cdot \frac{π}{180}$ radians

Step 2.5.3

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{2 + \sqrt{2}}{2} \cdot \frac{1}{2}} \cdot \frac{π}{180}$ radians

Step 2.5.4

Multiply $\frac{2 + \sqrt{2}}{2} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 2.5.4.1

Multiply $\frac{2 + \sqrt{2}}{2}$ by $\frac{1}{2}$ .

$\sqrt{\frac{2 + \sqrt{2}}{2 \cdot 2}} \cdot \frac{π}{180}$ radians

Step 2.5.4.2

Multiply $2$ by $2$ .

$\sqrt{\frac{2 + \sqrt{2}}{4}} \cdot \frac{π}{180}$ radians

Step 2.5.5

Rewrite $\sqrt{\frac{2 + \sqrt{2}}{4}}$ as $\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}}$ .

$\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}} \cdot \frac{π}{180}$ radians

Step 2.5.6

Simplify the denominator.

Tap for more steps...

Step 2.5.6.1

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{2^{2}}} \cdot \frac{π}{180}$ radians

Step 2.5.6.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\sqrt{2 + \sqrt{2}}}{2} \cdot \frac{π}{180}$ radians

Step 3

Multiply $\frac{\sqrt{2 + \sqrt{2}}}{2} \cdot \frac{π}{180}$ .

Tap for more steps...

Step 3.1

Multiply $\frac{\sqrt{2 + \sqrt{2}}}{2}$ by $\frac{π}{180}$ .

$\frac{\sqrt{2 + \sqrt{2}} π}{2 \cdot 180}$ radians

Step 3.2

Multiply $2$ by $180$ .

$\frac{\sqrt{2 + \sqrt{2}} π}{360}$ radians